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The Spatial Structure of Nuclei

Without a doubt there is some shell structure to nuclei. Furthermore neutrons and protons could have separate shells. Because of differences in the forces between the nucleons the neutron and proton shells might be spatially separated, but by not much, as shown below.

Schematic of Nuclear Shells

On the other hand, the protons and neutrons may be linked together in joint shells, as shown below

An examination of nuclear densities indicates that there is close packing of the nucleons. The shell capacities and the close packing of nucleons indicates that the spatial arrangement of a shell is three dimensional, like a sphere or cylinder, rather than two dimensional like a ring.

The values of the binding energies indicate that nucleons are arranged in pairs and wherever possible these pairs are combined into something like alpha particles.

Thus the first shell of the neutrons and the protons is simply an alpha particle at the center of the nucleus. The second shells might be effectively two alpha particles positioned antipodially.

Here an alpha particle is depicted as a white sphere. The alpha particles are located within two shells; the neutrons in the neutron shell, the protons in the proton shell.

The third shells might be four alpha particles arranged perhaps as a square ring or tetrahedron.

The fourth shells consist of seven alpha particles arranged who knows how. The fifth shells when complete consist of eleven alpha particles. The sixth when complete would consist of sixteen alpha particles and the seventh twenty two alpha particles. A shell of 16 or 22 alpha particles would approximate a spherical arrangement whereas the lower level shells would not. Thus the characteristics of the shells such as the incremental binding energies of the nucleons would approach regularity of the upper shells while being irregular for the lower shells.

Possible Subshells of the Neutron
Shell that Begins after 126 Neutrons

The relationships between the incremental binding energies of nucleons and the number of such nucleons are very regular for the larger nuclides, but not perfectly regular. This indicates that the spatial arrangement of nucleons in shells is symmetric but with some sort of subshell structure. Consider the following graph for the incremental binding energies of the neutrons in Uranium.

The sawtooth pattern comes from the formation of neutron pairs. There is a slight change in the pattern after 138 or 140 neutrons. The amplitude of the pair formation fluctuations changes and also the slope of the relationship. Thus there appears to be a subshell that has a capacity of 12 or 14 neutrons or equivalently 6 or 7 neutron pairs.

An examination of the relationship for the isotopes of Thorium, shown below, indicates that that the situation is more complex than a single subshell at 138 or 140 neutrons.

The change in the pattern at 126 neutrons is the filling of a neutron shell, what is called a magic number. The other more subtle changes occur at different numbers of neutrons other than the 138 or 140 seen in the case of Uranium.

The relationship for the isotopes of Protactinium (P=91) is similar to that of Thorium.

The change seems to come definitely at 140 neutrons rather than possibly 138 neutrons as for Uranium. A change at 140 neutrons indicates a subshell with a capacity of 14 neutrons.

The relationships for the elements Actinium and Radium are show together in the graph below. The correspondence is close and the change in the patterns come at 140 neutrons.

Thus there appears to be a subshell of the neutron shell that begins after 126 neutrons which has a capacity of 14 neutrons.

The next two elements Francium (P=89) and Radon (P=88) show no change at 140 neutrons. The correspondence of the two relationships however is notable.

Proton Shells

There exists no nuclides with proton numbers in the range considered above for neutrons. Instead the nuclides with 82 neutrons were investigatged. The incremental binding energies of the protons for these nuclides were computed and the results are shown below.

There appears to be no indication of a subshell for the nuclides considered. There are changes over the range of proton numbers but they are gradual and continuous.

(To be continued.)

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